Lubrication pressure and fractional viscous damping effects on the spring-block model of earthquakes

  • G. B. Tanekou
  • C. F. Fogang
  • R. Kengne
  • F. B. Pelap
Regular Article
  • 2 Downloads

Abstract.

We examine the dynamical behaviours of the “single mass-spring” model for earthquakes considering lubrication pressure effects on pre-existing faults and viscous fractional damping. The lubrication pressure supports a part of the load, thereby reducing the normal stress and the associated friction across the gap. During the co-seismic phase, all of the strain accumulated during the inter-seismic duration does not recover; a fraction of this strain remains as a result of viscous relaxation. Viscous damping friction makes it possible to study rocks at depth possessing visco-elastic behaviours. At increasing depths, rock deformation gradually transitions from brittle to ductile. The fractional derivative is based on the properties of rocks, including information about previous deformation events (i.e., the so-called memory effect). Increasing the fractional derivative can extend or delay the transition from stick-slip oscillation to a stable equilibrium state and even suppress it. For the single block model, the interactions of the introduced lubrication pressure and viscous damping are found to give rise to oscillation death, which corresponds to aseismic fault behaviour. Our result shows that the earthquake occurrence increases with increases in both the damping coefficient and the lubrication pressure. We have also revealed that the accumulation of large stresses can be controlled via artificial lubrication.

References

  1. 1.
    A. Namiki, T. Yamaguchi, I. Sumita, T. Suzuki, S. Ide, J. Geophys. Res. 119, 3169 (2014)ADSCrossRefGoogle Scholar
  2. 2.
    C.T. Sun, Y.P. Lu, Vibration Damping of Structural Elements (Prentice Hall PTR, New Jersey, USA, 1995)Google Scholar
  3. 3.
    C.M. Harris, A.G. Piersol, Harris’ Shock and Vibration Handbook, Fifth Edition (McGraw-Hill, New York, USA, 2002)Google Scholar
  4. 4.
    R.M. Christensen, Theory of Viscoelasticity: An Introduction (Academic Press, New York, 1971)Google Scholar
  5. 5.
    J.C. Simo, T.J.R. Hughes, Computational Inelasticity (Springer, New York, 1998)Google Scholar
  6. 6.
    D.I. Jones, Handbook of Viscoelastic Vibration Damping (J. Wiley & Son, New York, 2001)Google Scholar
  7. 7.
    A. Hanyga, J. Comput. Acoust. 11, 75 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D.L. Kohlstedt, B. Evans, S.J. Mackwell, J. Geophys. Res. 100, 17587 (1995)ADSCrossRefGoogle Scholar
  9. 9.
    R. Burgmann, G. Dresen, Annu. Rev. Earth Planet. Sci. 38, 531 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Y. Ito, K. Obara, T. Matsuzawa, T. Maeda, J. Geophys. Res. 114, B00A13 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    K. Obara, J. Geodyn. 52, 229 (2011)CrossRefGoogle Scholar
  12. 12.
    R.D. Hyndman, J. Geophys. Res. 118, 5530 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    T. Lay, H. Kanamori, C.J. Ammon, K.D. Koper, A.R. Hutko, L. Ye, H. Yue, T.M. Rushing, J. Geophys. Res. 117, B04311 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    D.A. Oleskevich, R.D. Hyndman, K. Wang, J. Geophys. Res. 104, 14965 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    B.W. Tichelaar, L.J. Ruff, J. Geophys. Res. 98, 2017 (1983)ADSCrossRefGoogle Scholar
  16. 16.
    K. Obara, Science 296, 1679 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    G. Rogers, H. Dragert, Science 300, 1942 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    A. Katsumata, N. Kamaya, Geophys. Res. Lett. 30, 20 (2003)ADSCrossRefGoogle Scholar
  19. 19.
    D.R. Shelly, G.G. Beroza, S. Ide, S. Nakamula, Nature 442, 188 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    H. Dragert, K. Wang, T.S. James, Science 292, 1525 (2001)ADSCrossRefGoogle Scholar
  21. 21.
    H. Hirose et al., Geophys. Res. Lett. 26, 3237 (1999)ADSCrossRefGoogle Scholar
  22. 22.
    A.M. Freed, R. Burgmann, E. Calais, J. Freymueller, Earth Planet. Sci. Lett. 252, 481 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    F.F. Pollitz, Earth Planet. Sci. Lett. 215, 89 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    R. Bürgmann, G. Dresen, Annu. Rev. Earth Planet. Sci. 36, 531 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    K. Wang, Y. Hu, J. He, Nature 484, 327 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    C.B. Rayleigh, J. Healy, J. Breadehoeft, Science 191, 1230 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    P. Bernard, “Qu’est ce qui fait trembler la terre?” (EDP Science, 2003)Google Scholar
  28. 28.
    M.W. Dongmo, L.Y. Kagho, F.B. Pelap, G.B. Tanekou, Y.L. Makenne, A. Fomethe, ISRN Geophys. 2014, 160378 (2014)CrossRefGoogle Scholar
  29. 29.
    F. Mulargia, A. Bizzarri, Sci. Rep. 4, 6100 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    R.H. Sibson, Fluid flow accompanying faulting: Field evidence and models, in Earthquake Prediction: An International Review, edited by M. Ewing Ser, D. Simpson, P. Richards, 4th ed. (American Geophysical Union, Washington, D.C., 1981) p. 593Google Scholar
  31. 31.
    R.H. Sibson, Pure Appl. Geophys. 124, 169 (1986)ADSCrossRefGoogle Scholar
  32. 32.
    C. Collettini, L. Chiaraluce, F. Pucci, M.R. Barchi, M. Cocco, J. Struct. Geol. 27, 937 (2005)ADSCrossRefGoogle Scholar
  33. 33.
    A. Nur, J. Booker, Science 175, 885 (1972)ADSCrossRefGoogle Scholar
  34. 34.
    A. Antonioli, M.E. Belardinelli, A. Bizzarri, K.S. Vogfjord, J. Geophys. Res. 111, B03302 (2006)ADSCrossRefGoogle Scholar
  35. 35.
    S.A. Miller, A. Nur, D.L. Olgaard, Geophys. Res. Lett. 23, 197 (1996)ADSCrossRefGoogle Scholar
  36. 36.
    S.A. Miller, C. Collettini, L. Chiaraluce, M. Cocco, M.R. Barchi, B.J.P. Kaus, Nature 427, 724 (2004)ADSCrossRefGoogle Scholar
  37. 37.
    T. Yamashita, Geophys. J. Int. 132, 661 (1998)CrossRefGoogle Scholar
  38. 38.
    S.A. Shapiro, R. Patzig, E. Rothert, J. Rindshwentner, Pure Appl. Geophys. 160, 1051 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    J. Byerlee, Geophys. Res. Lett. 17, 2109 (1990)ADSCrossRefGoogle Scholar
  40. 40.
    D. Lockner, J. Byerlee, Pure Appl. Geophys. 145, 717 (1995)ADSCrossRefGoogle Scholar
  41. 41.
    J.R. Rice, Fault stress states, pore pressure distributions, and the weakness of the San Andreas Fault, in Fault Mechanics and Transport Properties in Rocks (the Brace Volume), edited by B. Evans, T.-F. Wong (Academic, San Diego, 1992)Google Scholar
  42. 42.
    M.K. Hubbert, W.W. Rubey, Geol. Soc. Am. Bull. 70, 115 (1959)ADSCrossRefGoogle Scholar
  43. 43.
    A. Bizzarri, M. Cocco, J. Geophys. Res. 11, B05303 (2006)ADSGoogle Scholar
  44. 44.
    A. Bizzarri, M. Cocco, J. Geophys. Res. 111, B05304 (2006)ADSGoogle Scholar
  45. 45.
    A. Bizzarri, J. Geophys. Res. 117, B05304 (2012)ADSGoogle Scholar
  46. 46.
    A. Sommerfeld, Mechanics of Deformable Bodies (Academic, San Diego, CA, 1950)Google Scholar
  47. 47.
    E. Brodsky, Hiroo Kanamori, J. Geophys. Res. 106, 16357 (2001)ADSCrossRefGoogle Scholar
  48. 48.
    C.H. Scholz, Nature 391, 37 (1998)ADSCrossRefGoogle Scholar
  49. 49.
    E.A. Hetland, M. Simons, Geophys. J. Int. 181, 99 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    A. Bizzarri, Rev. Geophys. 49, RG3002 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    A. Bizzarri, Riv. Nuovo Cimento 37, 181 (2014)Google Scholar
  52. 52.
    R.H. Sibson, Nat. Phys. Sci. 243, 66 (1973)ADSCrossRefGoogle Scholar
  53. 53.
    Y.A. Fialko, J. Geophys. Res. 109, B01303 (2004)ADSGoogle Scholar
  54. 54.
    A. Bizzarri, M. Cocco, J. Geophys. Res. 111, B11302 (2006)ADSGoogle Scholar
  55. 55.
    J.R. Rice, J. Geophys. Res. 111, B05311 (2006)ADSCrossRefGoogle Scholar
  56. 56.
    A. Bizzarri, P. Spudich, J. Geophys. Res. 113, B05304 (2008)ADSCrossRefGoogle Scholar
  57. 57.
    A. Bizzarri, Geophys. Res. Lett. 36, L11304 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    A. Bizzarri, Earth Planet. Sci. Lett. 296, 144 (2010)ADSCrossRefGoogle Scholar
  59. 59.
    A. Bizzarri, J. Geophys. Res. 116, B02310 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    B. Erickson et al., Nonlinear Process. Geophys. 15, 1 (2008)ADSCrossRefGoogle Scholar
  61. 61.
    G.L. Vasconcelos, Phys. Rev. Lett. 76, 4865 (1996)ADSCrossRefGoogle Scholar
  62. 62.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)Google Scholar
  63. 63.
    V.T. Pham, C. Volos, S. Jafari, Z. Wei, X. Wang, Int. J. Bifurc. Chaos 24, 1450073 (2014)CrossRefGoogle Scholar
  64. 64.
    V.T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, Eur. Phys. J. ST 224, 1507 (2015)CrossRefGoogle Scholar
  65. 65.
    V.T. Pham, C. Volos, S. Jafari, T. Kapitaniak, Nonlinear Dyn. 87, 2001 (2017)CrossRefGoogle Scholar
  66. 66.
    I. Petráš, Fractional-Order Nonlinear Systems (Springer, Berlin, 2011)Google Scholar
  67. 67.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  68. 68.
    I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, New York, 2010)Google Scholar
  69. 69.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  70. 70.
    M. Hubbert, W. Rubey, Geol. Soc. Am. Bull. 70, 115 (1959)ADSCrossRefGoogle Scholar
  71. 71.
    Y. Mitsui, Change of Pore Fluid Pressure Versus Frictional Coefficient during Fault Slip, edited by Sebastiano D’Amico (2012)Google Scholar
  72. 72.
    J.H. Dieterich, B.D. Kilgore, Pure Appl. Geophys. 143, 283 (1994)ADSCrossRefGoogle Scholar
  73. 73.
    J.H. Dieterich, B.D. Kilgore, Tectonophysics 256, 219 (1996)ADSCrossRefGoogle Scholar
  74. 74.
    K. Terzaghi, Theoretical Soil Mechanics (John Wiley and Sons, NY, 1943) p. 510Google Scholar
  75. 75.
    W.F. Brace, R.J. Martin, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 5, 415 (1968)CrossRefGoogle Scholar
  76. 76.
    J. Handin, R.V. Hager, M. Friedman, J.N. Feather, Bull. Am. Assoc. Petrol. Geol. 47, 717 (1963)Google Scholar
  77. 77.
    M. de Sousa Vieira, Phys. Rev. A 46, 6288 (1992)ADSCrossRefGoogle Scholar
  78. 78.
    J. Dieterich, Pure Appl. Geophys. 116, 790 (1978)ADSCrossRefGoogle Scholar
  79. 79.
    A. Ruina, J. Geophys. Res. 88, 10359 (1983)ADSCrossRefGoogle Scholar
  80. 80.
    J.R. Rice, Pure Appl. Geophys. 121, 443 (1983)ADSCrossRefGoogle Scholar
  81. 81.
    A. Bizzarri, M. Cocco, Ann. Geophys. 48, 277 (2005)Google Scholar
  82. 82.
    A. Bizzarri, M. Cocco, J. Geophys. Res. 108, 2373 (2003)ADSCrossRefGoogle Scholar
  83. 83.
    J.M. Carlson, J.S. Langer, Phys. Rev. A 40, 6470 (1989)ADSMathSciNetCrossRefGoogle Scholar
  84. 84.
    J.M. Carlson, J.S. Langer, Phys. Rev. Lett. 62, 2632 (1989)ADSCrossRefGoogle Scholar
  85. 85.
    M. De Sousa Vieira, Phys. Rev. Lett. 82, 201 (1999)ADSCrossRefGoogle Scholar
  86. 86.
    S. Kostié et al., Nonlinear Proc. Geophys. 20, 857 (2013)ADSCrossRefGoogle Scholar
  87. 87.
    Y. Yamaguchi, H. Shimizu, Nonlinear Phenom. 11, 212 (1984)ADSCrossRefGoogle Scholar
  88. 88.
    D. Aronson, G. Ermentout, N. Kopell, Physica D 41, 403 (1990)ADSMathSciNetCrossRefGoogle Scholar
  89. 89.
    D. Reddy, A. Sen, G. Johnston, Phys. Rev. Lett. 80, 5109 (1998)ADSCrossRefGoogle Scholar
  90. 90.
    M. de Sousa Vieira, G.L. Vasconcelos, S.R. Nagel, Phys. Rev. E 47, R2221 (1993)ADSCrossRefGoogle Scholar
  91. 91.
    G.L. Vasconcelos, M. de Sousa Vieira, S.R. Nagel, Physica A 191, 69 (1992)ADSCrossRefGoogle Scholar
  92. 92.
    B. Gutenberg, C.F. Richter, Seismicity of the Earth (Hafner, New York, 1965)Google Scholar
  93. 93.
    C.F. Richter, Bull. Seism. Soc. Am. 25, 1 (1935)Google Scholar
  94. 94.
    B. Gutenberg, Bull. Seism. Soc. Am. 35, 3 (1945)Google Scholar
  95. 95.
    B. Gutenberg, C.F. Richter, Ann. Geofis. 9, 7 (1956)Google Scholar
  96. 96.
    H. Kanamori, J. Geophys. Res. 82, 2981 (1977)ADSCrossRefGoogle Scholar
  97. 97.
    T.C. Hanks, H.J. Kanamori, Geophys. Res. 84, 23 (1979)Google Scholar
  98. 98.
    T. Utsu, Statistical features of seismology, in International Handbook of Earthquake and Engineering Seismology, Part A, edited by W.H.K. Lee, H. Kanamori, P.C. Jennings, C. Kisslinger (Academic Press, 2002) pp. 719--732Google Scholar
  99. 99.
    H. Kanamori, D. Hadley, Pure Appl. Geophys. 113, 32 (1975)CrossRefGoogle Scholar
  100. 100.
    H. Kanamori, J. Mori, B. Sturtevant, D. Anderson, T. Heaton, Shock Waves Int. J. 2, 89 (1992)ADSCrossRefGoogle Scholar
  101. 101.
    K. Aki, Bull. Earth Res. Inst. (Tokyo Univ.) 44, 73 (1966)Google Scholar
  102. 102.
    F.B. Pelap, L.Y. Kagho, C.F. Fogang, Chaos Solitons Fractals 87, 71 (2016)ADSMathSciNetCrossRefGoogle Scholar
  103. 103.
    E.G. Daub et al., Geophys. Res. Lett. 38, L10301 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • G. B. Tanekou
    • 1
  • C. F. Fogang
    • 1
  • R. Kengne
    • 2
  • F. B. Pelap
    • 1
  1. 1.UR de Mécanique et de Modélisation des Systèmes Physiques (UR-2MSP), UFR/DSSTUniversité de DschangDschangCameroon
  2. 2.UR de Matière Condensée Electronique et traitement du signal (UR-MACETS)UFR/DSST, Université de DschangDschangCameroon

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